Selective game versions of countable tightness with bounded finite selections

TL;DR

This paper explores variations of a topological game related to countable tightness, where the second player chooses finitely many points per inning with fixed or increasing limits, revealing the distinctness of these game variants.

Contribution

It introduces and analyzes new game variants allowing finitely many points per inning with fixed or increasing limits, showing their distinctness and implications for countable tightness.

Findings

All fixed-limit game variants are distinct.

A new game variant with increasing points per inning is introduced.

Abstract

For a topological space $X$ and a point $x \in X$, consider the following game -- related to the property of $X$ being countably tight at $x$. In each inning $n\in\omega$, the first player chooses a set $A_n$ that clusters at $x$, and then the second player picks a point $a_n\in A_n$; the second player is the winner if and only if $x\in\overline{\{a_n:n\in\omega\}}$. In this work, we study variations of this game in which the second player is allowed to choose finitely many points per inning rather than one, but in which the number of points they are allowed to choose in each inning has been fixed in advance. Surprisingly, if the number of points allowed per inning is the same throughout the play, then all of the games obtained in this fashion are distinct. We also show that a new game is obtained if the number of points the second player is allowed to pick increases at each inning.